Question

Find the exact value of sin240°.

Answer 1

`\sin240^\circ=\sin(180-240)^\circ=\sin(-60^\circ)=-\sin60^\circ=-\frac{\sqrt3}2`

using the fact that (1)
`\sin x=\sin(\pi -x)` (where
`x` is in radians; replace with
`180^\circ` to convert to degrees) and (2)
`\sin(-x)=-\sin x` for all
`x`. Then
`\sin\frac\pi3=\sin60^\circ` is a known value.

Answer 2

`-(√(3))/(2)`

Explanation:

We need to find the value of 240°

To find the value, first we need to understand in which quadrant 240° lies.

We know that there are four quadrants.

First quadrant lies from
`(0,\:(\pi )/(2))` in radians or 0 to 90° in degrees.

Second quadrant lies from
`((\pi )/(2),\:\pi)` in radians or 90° to 180° in degrees.

Third quadrant lies from
`(\pi,\:(3\pi )/(2))` in radians or 180° to 270° in degrees.

Fourth quadrant lies from
`((3\pi )/(2),\:}2\pi )` in radians or 270° to 360° in degrees.

Clearly we can see that 240° lies in the third quadrant, and in the third quadrant sin theta is negative.

Now,
`\text{sin\:240}^(\circ)=\text{sin\:(180+60)}^(\circ)`

We know that
`\text{sin}(\pi +x)=-\text{sin}x`

So,
`\text{sin\:240}^(\circ)=-\text{sin\:60}^(\circ)`

Hence,
`\text{sin\:240}^(\circ)=-(√(3))/(2)`